Hongen Jia

We propose a fully decoupled first‐order numerical scheme for the numerical solution of the thermally coupled magnetohydrodynamics (MHD) equations with variable density in this paper. First, we construct an equivalent reformulation of the governing equations. On this basis, a fully decoupled time‐discretization scheme is developed by combining the Gauge‐Uzawa method and the auxiliary variable approach, which decouples velocity and pressure and efficiently handles the nonlinear convective terms. Compared with traditional pressure correction methods, the Gauge‐Uzawa method achieves direct one‐step velocity–pressure decoupling without additional pressure correction steps, thus ensuring efficiency, maintaining pressure accuracy, and improving numerical stability and precision. Second, the unconditional energy stability of the scheme is rigorously proved. Furthermore, we overcome the difficulties caused by the strong nonlinearity brought by variable density and the strong coupling among thermal‐fluid‐magnetic multiphysics fields, and complete the first rigorous error analysis of the scheme. Eventually, the validity of the method is confirmed by a sequence of numerical tests.

This paper proposes an efficient deep encoder–decoder ConvLSTM neural network for solving time-dependent differential partial equations. The encoder is composed of convolution layers, and the decoder is composed of transposed convolution layers, where the image is converted from low frequency to high frequency, and then back to low frequency in order to extract the spatial features of the partial differential equation. To accurately capture temporal information of time-dependent partial differential equations, ConvLSTM is configured in the encoder and decoder to solve the long-term evolution of dynamic systems. Furthermore, skip connections are added between the convolutional layer and the corresponding transposed convolutional layer, which not only overcomes the problem of gradient disappearance but also ensures that the underlying features are learned, enhancing the accuracy and efficiency of our network. Finally, the proposed network has been used to solve heat equations, wave equations, 2D Burgers equations, and high-order equations, and the results show that the present neural network has much better performance in terms of accuracy and efficiency.

In this paper, an accurate fractional physical information neural network with an adaptive learning rate (adaptive-fPINN-PQI) was first proposed for solving fractional partial differential equations. First, piecewise quadratic interpolation (PQI) in the sense of the Hadamard finite-part integral was introduced in the neural network to discretize the time-fractional derivative in the Caputo sense. Second, the adaptive learning rate residual network was constructed to keep the network from being stuck in the locally optimal solution, which automatically adjusts the weights of different loss terms, significantly balancing their gradients. Additionally, different from the traditional physical information neural networks, this neural network employs a new composite activation function based on the principle of Fourier transform instead of a single activation function, which significantly enhances the network's accuracy. Finally, numerous time-fractional diffusion and time-fractional phase-field equations were solved using the proposed adaptive-fPINN-PQI to demonstrate its high precision and efficiency.

In this study, we construct an error estimate for a fully discrete finite element scheme that satisfies the criteria of unconditional energy stability, as suggested in ^[1]. Our theoretical findings, in more detail, demonstrate that this system has second-order accuracy in both space and time. Additionally, we offer a powerful space and time adaptable approach for solving the Cahn-Hilliard problem numerically based on the posterior error estimation. The major goal of this technique is to successfully lower the calculated cost by controlling the mesh size using a Superconvergent Cluster Recovery (SCR) approach in accordance with the error estimation. To demonstrate the effectiveness and stability of the suggested SCR-based algorithm, numerical results are provided.

In this paper, a new decoupling method is proposed to solve a nematic liquid crystal flow with stretching effect. In the finite element discrete framework, the director vector is calculated by introducing a new auxiliary variable w, and the velocity vector and scalar pressure are decoupled by a nonincremental pressure-correction projection method. Then, the energy dissipation law and unconditional energy stability of the resulting system are given. Finally, some numerical examples are given to verify the effects of various parameters on the singularity annihilation, stability and accuracy in space and time.